376 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIFPIC INTEGRALS. 
dx / V 
Hence 
and if, as before, we write M for 1 — msin^<p, or 1 — we shall have 
d<p 
(294.) 
o-_x/ M^/r 
Differentiating this expression on the hypothesis that i and (p are constant, while e is 
r 
variable, we shall have 
dcr 1 /-dxC^ 
do- 1 /-dx|r 2 A<p , dip C 
d7=2V m 7I+ e LJ„ Jo 
Multiplying this equation by and recollecting that — 2e^l — vve shall have 
dip 
m^T 
dx 
. . . (295.) 
Vx dir 
e de 
( dip 1 2x^2 d^ dip 
Mv/T+^Jo 
M'/l ■ e" 
But (see FIymer’s Integral Calculus, p. 195) 
Ma/I' 
. . . (296.) 
J: 
dp 
Introducing this value into the preceding equation, the coefficient of | 
vanish, and we shall have 
A^x dcr 
= will 
. . i/x 
Dividing by and integrating on the hypothesis that p and i are constant, 
r . n 
E=-(^’)J;^+?J;dfv/i ( 298 -) 
(7 = 
or as 
e\/ x.—^y (1— e‘'^)(e^ — f), we shall have 
'dpj^/l 
de 
*2 dp 
fde 
J e^V l-e‘-^' 
f constant. . . . (299.) 
{l—e^)[e^—i^) [_Jo a/iJ^ 
M^e must recollect that the definite integrals within the brackets are functions, not 
TT 
of p, but of F, 0, and They are therefore constants. 
It is not a little remarkable that the coefficients of the dejinite elliptic integrals are 
themselves also elliptic integrals of the first and second orders. To show this, assume 
e^=cos^0d-i^ sin^0 (300.) 
Therefore 1 sin®0, and cos^d ; we have also ede= — sin0 cos0d(f. 
d0 Cd0 
Hence, if I -/ sin’fe -J- 
'v/ 1 sin^^ J a/ J’ 
and 
/- sin 0 COS 0 
^ ^ sin^d' 
. (301.) 
. (302.) 
